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<title>Elektrisches Potential</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Elektrisches Potential</span></h1>

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<th colspan="2" style="background:#ABCDEF; color:inherit;"><a href="Physikalische_Gr%C3%B6%C3%9Fe" title="Physikalische Größe">Physikalische Größe</a>

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<td style="width:130px;">Name

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<td><b>elektrisches Potential</b>

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<td><a href="Formelzeichen" title="Formelzeichen">Formelzeichen</a>

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<annotation encoding="application/x-tex">{\displaystyle \varphi ,\,\phi ,\,\Phi ,\,V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fd47d93c2dab1602d2ae2db0cd6442b791e132c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.634ex; height:2.676ex;" alt="{\displaystyle \varphi ,\,\phi ,\,\Phi ,\,V}" loading="lazy"></span></i>

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<th><a href="Gr%C3%B6%C3%9Fensystem" title="Größensystem">Größen-</a> und<br><a href="Einheitensystem" title="Einheitensystem">Einheitensystem</a>

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<th><a href="Ma%C3%9Feinheit" title="Maßeinheit">Einheit</a>

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<th><a href="Dimension_(Gr%C3%B6%C3%9Fensystem)" title="Dimension (Größensystem)">Dimension</a>

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<td><a href="Gau%C3%9Fsches_Einheitensystem" title="Gaußsches Einheitensystem">Gauß</a>, <a href="Elektrostatisches_Einheitensystem" title="Elektrostatisches Einheitensystem">esE</a> (<a href="CGS-Einheitensystem" title="CGS-Einheitensystem">cgs</a>)

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<td><a href="Statvolt" title="Statvolt">Statvolt</a> (statV)

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<td><a href="Elektromagnetisches_Einheitensystem" title="Elektromagnetisches Einheitensystem">emE</a> (<a href="CGS-Einheitensystem" title="CGS-Einheitensystem">cgs</a>)

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<td><a href="Abvolt" title="Abvolt">Abvolt</a> (abV)

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<td><a href="Planck-Einheiten" title="Planck-Einheiten">Planck</a>

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<td>1

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<p>Das <b>elektrische Potential</b> oder <b>Coulomb-Potential</b> (auch geschrieben als <b>Potenzial</b>) ist eine <a href="Physikalische_Gr%C3%B6%C3%9Fe" title="Physikalische Größe">physikalische Größe</a> in der <a href="Elektrodynamik" title="Elektrodynamik">klassischen Elektrodynamik</a>. Es beschreibt die Fähigkeit eines <a href="Elektrische_Feldst%C3%A4rke" title="Elektrische Feldstärke">elektrischen Feldes</a>, <a href="Arbeit_(Physik)" title="Arbeit (Physik)">Arbeit</a> an einer <a href="Elektrische_Ladung" title="Elektrische Ladung">elektrischen Ladung</a> zu verrichten. Die <a href="Internationales_Einheitensystem" title="Internationales Einheitensystem">SI-Einheit</a> für das elektrische Potential ist <a href="Volt" title="Volt">Volt</a>. Das Formelzeichen ist meistens ein kleines oder großes <a href="Phi" title="Phi">Phi</a>, also <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

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<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aed80a2011a3912b028ba32a52dfa57165455f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" loading="lazy"></span>, oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>.

</p><p>Die Differenz der Potentiale zwischen zwei Punkten wird als <a href="Elektrische_Spannung" title="Elektrische Spannung">elektrische Spannung</a> bezeichnet.

</p><p>Ein gegebenes elektrisches Feld ordnet jedem Punkt im Raum ein Potential zu, das bis auf eine Konstante eindeutig ist. Wenn das Potential im gesamten Raum betrachtet wird, spricht man auch von einem <a href="Potentialfeld" class="mw-redirect" title="Potentialfeld">Potentialfeld</a>.

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<div class="mw-heading mw-heading2"><h2 id="Anschauliche_Erklärung"><span id="Anschauliche_Erkl.C3.A4rung"></span>Anschauliche Erklärung</h2></div>

<p>Das elektrische Potential beschreibt die Fähigkeit eines elektrischen Feldes, <a href="Arbeit_(Physik)" title="Arbeit (Physik)">Arbeit</a> an einer elektrischen Ladung zu verrichten. Wenn sich eine Probeladung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span> durch ein elektrisches Feld bewegt, wirkt auf sie die <a href="Coulombkraft" class="mw-redirect" title="Coulombkraft">Coulombkraft</a> und es wird Arbeit an ihr geleistet. Sie erhält dadurch <a href="Potentielle_Energie" title="Potentielle Energie">potentielle Energie</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\rm {pot}}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{\rm {pot}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c272c93a1e4f00aae6a0a26090b12cdda23a9e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.802ex; height:2.843ex;" alt="{\displaystyle W_{\rm {pot}}}" loading="lazy"></span>. Die Stärke der Coulombkraft und damit die Größe der zugeführten potentiellen Energie hängt von der Größe der Ladung ab. Um eine allgemeinere Darstellung der potentiellen Energie unabhängig von der Ladung zu erhalten, wird das elektrische Potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> eingeführt, indem die potentielle Energie durch die Ladung geteilt wird:

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<annotation encoding="application/x-tex">{\displaystyle \varphi ={\frac {W_{\text{pot}}}{q}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d3ed3988537476a92798e1ce1043d90440581ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.256ex; height:6.009ex;" alt="{\displaystyle \varphi ={\frac {W_{\text{pot}}}{q}}}" loading="lazy"></span></dd></dl>

<p>Das Potential gibt damit an, wie viel potentielle Energie eine Ladung pro Ladungseinheit im elektrischen Feld hat. Wenn das elektrische Feld sich nicht mit der Zeit verändert (siehe <a href="Elektrostatik" title="Elektrostatik">Elektrostatik</a>), kann man das elektrische Potential als eine Art „Energie pro Ladung“ betrachten. Wenn das elektrische Feld sich jedoch im Laufe der Zeit ändert (siehe <a href="Elektrodynamik" title="Elektrodynamik">Elektrodynamik</a>), muss die Definition des elektrischen Potentials angepasst werden.

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<div class="mw-heading mw-heading2"><h2 id="In_der_Elektrostatik">In der Elektrostatik</h2></div>

<p>Kennt man die potentielle Energie <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\text{pot}}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{\text{pot}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dff1adfda917410d56c914cadb43a015a349ddf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.802ex; height:2.843ex;" alt="{\displaystyle W_{\text{pot}}}" loading="lazy"></span> für eine unbewegte <a href="Punktladung" title="Punktladung">Punktladung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span> im gesamten Raum, berechnet sich das elektrische Potential durch

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<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}})={\frac {W_{\text{pot}}({\vec {r}})}{q}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/344ad32459a49cf253d73603d4bf3111271a1045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.321ex; height:6.176ex;" alt="{\displaystyle \varphi ({\vec {r}})={\frac {W_{\text{pot}}({\vec {r}})}{q}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Elektrisches_Potential_einer_Punktladung">Elektrisches Potential einer Punktladung</h3></div>

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<div class="thumbcaption" style="clear:both;text-align:left;">Elektrisches Potential einer positiven bzw. negativen Punktladung. Die Stärke des Potentials wird durch den Farbverlauf von Magenta (+) über gelb (0) zu blau (-) angegeben. Die ringförmigen Linien geben die <a href="%C3%84quipotentialfl%C3%A4che" title="Äquipotentialfläche">Äquipotentialflächen</a> an, die anderen Linien, das <a href="Elektrische_Feldst%C3%A4rke" title="Elektrische Feldstärke">elektrische Feld</a>.</div></div></div>

<p>Das elektrische Potential einer unbewegten <a href="Punktladung" title="Punktladung">Punktladung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span>, auch <a href="Coulombsches_Gesetz#Coulomb-Potential" title="Coulombsches Gesetz">Coulomb-Potential</a> genannt, ist im <a href="SI-Einheitensystem" class="mw-redirect" title="SI-Einheitensystem">SI-Einheitensystem</a> gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}})={\frac {q}{4\pi \,\varepsilon _{0}\,\left|{\vec {r}}\right|}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b10f1d7769fc57cdd146aed2b3cae6286cff32d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.797ex; height:5.676ex;" alt="{\displaystyle \varphi ({\vec {r}})={\frac {q}{4\pi \,\varepsilon _{0}\,\left|{\vec {r}}\right|}}}" loading="lazy"></span></dd></dl>

<p>Dabei bezeichnet

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span> die <a href="Elektrische_Ladung" title="Elektrische Ladung">elektrische Ladung</a></dd>

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"></span> die <a href="Elektrische_Feldkonstante" title="Elektrische Feldkonstante">elektrische Feldkonstante</a></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"></span> die Position des betrachteten Punktes relativ zur Punktladung.</dd></dl>

<p>Im <a href="Heaviside-Lorentz-Einheitensystem" title="Heaviside-Lorentz-Einheitensystem">Heaviside-Lorentz-Einheitensystem</a> gilt wegen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}=1}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}=1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fed767ce6400c809b15275efc75fccb532dc14e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.399ex; height:2.509ex;" alt="{\displaystyle \varepsilon _{0}=1}" loading="lazy"></span> vereinfacht

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<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}})={\frac {q}{4\pi \,\left|{\vec {r}}\right|}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9337399de9bfb2cb887844dc66a6d6ab4f6a592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.273ex; height:5.676ex;" alt="{\displaystyle \varphi ({\vec {r}})={\frac {q}{4\pi \,\left|{\vec {r}}\right|}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Elektrisches_Potential_eines_beliebigen_statischen_Feldes">Elektrisches Potential eines beliebigen statischen Feldes</h3></div>

<p>Statische <a href="Elektrisches_Feld" title="Elektrisches Feld">elektrische Felder</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bc18ae485a72f148e85ccbeff2b3dcdd4f5f3f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\vec {E}}}" loading="lazy"></span> sind <a href="Wirbelfreies_Vektorfeld" class="mw-redirect" title="Wirbelfreies Vektorfeld">wirbelfrei</a>, sie können deshalb als <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradient</a> eines <a href="Skalarfeld" title="Skalarfeld">Skalarfeldes</a> dargestellt werden (siehe <a href="Gradientenfeld" title="Gradientenfeld">Gradientenfeld</a>). Das negative Skalarfeld wird dabei als elektrisches Potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> bezeichnet.

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<annotation encoding="application/x-tex">{\displaystyle -{\vec {\nabla }}\varphi ={\vec {E}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/157f8e50bc8ec520c35f8862ca11514558389d26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.138ex; height:3.343ex;" alt="{\displaystyle -{\vec {\nabla }}\varphi ={\vec {E}}}" loading="lazy"></span></dd></dl>

<p>Ist das <a href="Elektrisches_Feld" title="Elektrisches Feld">elektrische Feld</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bc18ae485a72f148e85ccbeff2b3dcdd4f5f3f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\vec {E}}}" loading="lazy"></span> bekannt, so lässt sich das Potential am Punkt mit dem <a href="Ortsvektor" title="Ortsvektor">Ortsvektor</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"></span>, ausgehend von einem Nullpotential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ({\vec {r}}_{0})=0}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}}_{0})=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e623da1efadfdb4b9f7a579ef0719d4bc1c8630a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.868ex; height:2.843ex;" alt="{\displaystyle \varphi ({\vec {r}}_{0})=0}" loading="lazy"></span> im Ort <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{0}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef4a62186256445a64da937f624fd1d892fad9ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{0}}" loading="lazy"></span>, durch ein <a href="Kurvenintegral" title="Kurvenintegral">Kurvenintegral</a> berechnen:

</p>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}})=-\int _{{\vec {r}}_{0}}^{\vec {r}}{\vec {E}}\cdot \mathrm {d} {\vec {s}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f6ac259f9464e4397ff5e2f430feffcfbcf9c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.803ex; height:7.009ex;" alt="{\displaystyle \varphi ({\vec {r}})=-\int _{{\vec {r}}_{0}}^{\vec {r}}{\vec {E}}\cdot \mathrm {d} {\vec {s}}}" loading="lazy"></span></dd></dl>

<p>Üblicherweise wird <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\infty )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2da3d433559cdbd3342fd0167a75e3bc82d176da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.653ex; height:2.843ex;" alt="{\displaystyle \varphi (\infty )}" loading="lazy"></span> als Nullpotential gewählt. Daraus folgt:

</p>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}})=\int _{\vec {r}}^{\infty }{\vec {E}}\cdot \mathrm {d} {\vec {s}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2846f10406288d89aa13841fd6936d21a892557c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.348ex; height:5.843ex;" alt="{\displaystyle \varphi ({\vec {r}})=\int _{\vec {r}}^{\infty }{\vec {E}}\cdot \mathrm {d} {\vec {s}}}" loading="lazy"></span></dd></dl>

<p>Im Innern eines <a href="Leiter_(Physik)#Elektrischer_Leiter" title="Leiter (Physik)">Leiters</a> ist das elektrische Potential wegen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}=0}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ef9dbb2d18828af64833581b3e004f0e7882bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.843ex;" alt="{\displaystyle {\vec {E}}=0}" loading="lazy"></span> damit konstant.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Für eine bekannte <a href="Ladungsdichte" title="Ladungsdichte">Ladungsverteilung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ({\vec {r}})}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ({\vec {r}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c80c262b5476b043478d88adb73498aac8b80de9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.234ex; height:2.843ex;" alt="{\displaystyle \rho ({\vec {r}})}" loading="lazy"></span> gilt:

</p>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}})={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho ({\vec {r}}^{\prime })}{\left|{\vec {r}}-{\vec {r}}^{\prime }\right|}}\,\mathrm {d} {\vec {r}}^{\prime }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e0df5896086a5f8f78580342b2794163a96f862.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.776ex; height:7.176ex;" alt="{\displaystyle \varphi ({\vec {r}})={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho ({\vec {r}}^{\prime })}{\left|{\vec {r}}-{\vec {r}}^{\prime }\right|}}\,\mathrm {d} {\vec {r}}^{\prime }}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Poisson-Gleichung">Poisson-Gleichung</h3></div>

<p>Für eine kontinuierliche Ladungsverteilung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> gilt die <a href="Poisson-Gleichung#Elektrostatik" title="Poisson-Gleichung">Poisson-Gleichung</a>:

</p>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \varphi =-{\frac {\rho }{\varepsilon _{0}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dd578162744e58d2d6451bb4381ffa7922255c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.336ex; height:5.176ex;" alt="{\displaystyle \Delta \varphi =-{\frac {\rho }{\varepsilon _{0}}}}" loading="lazy"></span></dd></dl>

<p>Speziell für den leeren Raum ergibt sich aus der Poisson-Gleichung mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ba6310b27df5f9c9b0b1732e08cce27b99d68cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho =0}" loading="lazy"></span> die <a href="Laplace-Gleichung" title="Laplace-Gleichung">Laplace-Gleichung</a>

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \varphi =0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae202f10d6fab9eea1024f2b824210a99d3d64d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.717ex; height:2.676ex;" alt="{\displaystyle \Delta \varphi =0}" loading="lazy"></span>.</dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> ist damit eine <a href="Harmonische_Funktion" title="Harmonische Funktion">harmonische Funktion</a>.

</p><p>Dabei bezeichnet

</p>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65eeb56775c5bb9a9645ca773c8b8f744ca6ce0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.843ex;" alt="{\displaystyle {\vec {\nabla }}}" loading="lazy"></span> den <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</a></dd>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta ={\vec {\nabla }}^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ac24b57cfedab2428e745f5fff2a29d7af20978.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.024ex; height:3.343ex;" alt="{\displaystyle \Delta ={\vec {\nabla }}^{2}}" loading="lazy"></span> den <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a></dd>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"></span> die <a href="Elektrische_Feldkonstante" title="Elektrische Feldkonstante">elektrische Feldkonstante</a>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="In_der_Elektrodynamik">In der Elektrodynamik</h2></div>

<p>Dynamische elektrische Felder sind nicht <a href="Wirbelfrei" class="mw-redirect" title="Wirbelfrei">wirbelfrei</a>, weil nach dem <a href="Induktionsgesetz" class="mw-redirect" title="Induktionsgesetz">Induktionsgesetz</a>

</p>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bddc706af01b0f56cdd6d4f2e1edd8b38149b708.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.377ex; height:6.176ex;" alt="{\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}" loading="lazy"></span></dd></dl>

<p>gilt, und können deshalb nicht als <a href="Gradientenfeld" title="Gradientenfeld">Gradientenfelder</a> dargestellt werden. Wirbelfrei ist hingegen der Ausdruck

</p>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}\times \left({\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}\right)=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1a9c80bbbe4f7111ef644d61c864a3c5bcc6a10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.232ex; height:7.509ex;" alt="{\displaystyle {\vec {\nabla }}\times \left({\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}\right)=0}" loading="lazy"></span>.</dd></dl>

<p>Dabei bezeichnet

</p>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {B}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83ae7d80cab55b606de217162280b2279142bbb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}}" loading="lazy"></span> die <a href="Magnetische_Flussdichte" title="Magnetische Flussdichte">magnetische Flussdichte</a>,</dd>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/391292ffadc65b0cde3e96f23afcdb811619dd95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:3.009ex;" alt="{\displaystyle {\vec {A}}}" loading="lazy"></span> das <a href="Magnetisches_Vektorpotential" title="Magnetisches Vektorpotential">magnetische Vektorpotential</a> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {B}}=\nabla \times {\vec {A}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {B}}=\nabla \times {\vec {A}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e7407e07bdb01a1048a72518d136819c32434f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.382ex; height:3.009ex;" alt="{\displaystyle {\vec {B}}=\nabla \times {\vec {A}}}" loading="lazy"></span>.</dd></dl>

<p>Dieses wirbelfreie <a href="Vektorfeld" title="Vektorfeld">Vektorfeld</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}}">

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11bf83a132ccd95add5dbce462c74e230133a7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.513ex; height:6.176ex;" alt="{\displaystyle {\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}}" loading="lazy"></span> ist mit dem elektrischen Potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> als Gradientenfeld darstellbar:

</p>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\vec {\nabla }}\varphi ={\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09bf4010eff411f9ca275130dc778c762d8eb016.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.876ex; height:6.176ex;" alt="{\displaystyle -{\vec {\nabla }}\varphi ={\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}}" loading="lazy"></span></dd></dl>

<p>Umgekehrt lässt sich das Potential an einem Ort <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"></span>, ausgehend von einem Nullpotential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ({\vec {r}}_{0})=0}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}}_{0})=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e623da1efadfdb4b9f7a579ef0719d4bc1c8630a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.868ex; height:2.843ex;" alt="{\displaystyle \varphi ({\vec {r}}_{0})=0}" loading="lazy"></span> in einem beliebig gewählten Ort <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{0}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef4a62186256445a64da937f624fd1d892fad9ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{0}}" loading="lazy"></span>, durch ein <a href="Kurvenintegral" title="Kurvenintegral">Kurvenintegral</a> bestimmen:

</p>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

</msubsup>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}},t)=-\int _{{\vec {r}}_{0}}^{\vec {r}}\left({\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}\right)\cdot \mathrm {d} {\vec {s}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5332232e1736dba6dfaffd0ccb17c43b279ec364.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.095ex; height:7.509ex;" alt="{\displaystyle \varphi ({\vec {r}},t)=-\int _{{\vec {r}}_{0}}^{\vec {r}}\left({\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}\right)\cdot \mathrm {d} {\vec {s}}}" loading="lazy"></span></dd></dl>

<p>Mit der üblichen Wahl von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (\infty )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2da3d433559cdbd3342fd0167a75e3bc82d176da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.653ex; height:2.843ex;" alt="{\displaystyle \varphi (\infty )}" loading="lazy"></span> als Nullpotential folgt:

</p>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}},t)=\int _{\vec {r}}^{\infty }\left({\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}\right)\cdot \mathrm {d} {\vec {s}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b0fe3a43f58df635d6052234fb9bb72ebbce9c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.64ex; height:7.509ex;" alt="{\displaystyle \varphi ({\vec {r}},t)=\int _{\vec {r}}^{\infty }\left({\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}\right)\cdot \mathrm {d} {\vec {s}}}" loading="lazy"></span></dd></dl>

<p>Für eine bekannte <a href="Ladungsdichte" title="Ladungsdichte">Ladungsverteilung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ({\vec {r}})}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ({\vec {r}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c80c262b5476b043478d88adb73498aac8b80de9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.234ex; height:2.843ex;" alt="{\displaystyle \rho ({\vec {r}})}" loading="lazy"></span> mit der <a href="Coulomb-Eichung" title="Coulomb-Eichung">Coulomb-Eichung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\nabla }}{\vec {A}}=0}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}{\vec {A}}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb881d75ba8fecf044be59e28ec9304542489c62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.94ex; height:3.009ex;" alt="{\displaystyle {\vec {\nabla }}{\vec {A}}=0}" loading="lazy"></span> gilt wie in der Elektrostatik:

</p>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ({\vec {r}},t)={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho ({\vec {r}}^{\prime },t)}{\left|{\vec {r}}-{\vec {r}}^{\prime }\right|}}\,\mathrm {d} {\vec {r}}^{\prime }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40825b8330de3965c37ec7dfa5769bfdfe126ab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.649ex; height:7.176ex;" alt="{\displaystyle \varphi ({\vec {r}},t)={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho ({\vec {r}}^{\prime },t)}{\left|{\vec {r}}-{\vec {r}}^{\prime }\right|}}\,\mathrm {d} {\vec {r}}^{\prime }}" loading="lazy"></span></dd></dl>

<p>Für stationäre Felder gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {\vec {A}}}{\partial t}}=0}">

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\vec {A}}}{\partial t}}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66993f6c55a4e74013ada9dab84ad284a8215b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.158ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial {\vec {A}}}{\partial t}}=0}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi }{\partial t}}=0}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20b8c1bb1ce743fb439edd5cec3867a0e9977fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.935ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}=0}" loading="lazy"></span>, sodass die dynamischen Gleichungen wieder in die Gleichungen für statische Felder übergehen.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Poisson-Gleichung_2">Poisson-Gleichung</h3></div>

Mit der <a href="Lorenz-Eichung" title="Lorenz-Eichung">Lorenz-Eichung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\nabla }}{\vec {A}}=-{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}{\vec {A}}=-{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71ae4efa2632ac79da3d422fc6b0adf26a09c346.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.157ex; height:5.843ex;" alt="{\displaystyle {\vec {\nabla }}{\vec {A}}=-{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}}" loading="lazy"></span> folgt für eine kontinuierliche <a href="Ladungsverteilung" class="mw-redirect" title="Ladungsverteilung">Ladungsverteilung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> die <a href="Poisson-Gleichung#Elektrostatik" title="Poisson-Gleichung">Poisson-Gleichung</a>:

</p>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df466ce63bed29b36208199b28fe1e0e9fba2c33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.828ex; height:6.009ex;" alt="{\displaystyle \Delta \varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}" loading="lazy"></span></dd></dl>

<p>Mit der <a href="Coulomb-Eichung" title="Coulomb-Eichung">Coulomb-Eichung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\nabla }}{\vec {A}}=0}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}{\vec {A}}=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb881d75ba8fecf044be59e28ec9304542489c62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.94ex; height:3.009ex;" alt="{\displaystyle {\vec {\nabla }}{\vec {A}}=0}" loading="lazy"></span> folgt hingegen

</p>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \varphi =-{\frac {\rho }{\varepsilon _{0}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dd578162744e58d2d6451bb4381ffa7922255c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.336ex; height:5.176ex;" alt="{\displaystyle \Delta \varphi =-{\frac {\rho }{\varepsilon _{0}}}}" loading="lazy"></span></dd></dl>

<p>Dabei bezeichnet

</p>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta ={\vec {\nabla }}^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ac24b57cfedab2428e745f5fff2a29d7af20978.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.024ex; height:3.343ex;" alt="{\displaystyle \Delta ={\vec {\nabla }}^{2}}" loading="lazy"></span>den <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a></dd>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"></span> die <a href="Elektrische_Feldkonstante" title="Elektrische Feldkonstante">elektrische Feldkonstante</a></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> die <a href="Lichtgeschwindigkeit" title="Lichtgeschwindigkeit">Lichtgeschwindigkeit</a>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Eichtransformation">Eichtransformation</h3></div>

<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Eichtransformation" title="Eichtransformation">Eichtransformation</a></i></div>

<p>In der Elektrostatik konnte das Potential bereits durch die freie Wahl des Nullpotentials um eine beliebige Konstante verschoben werden. In der Elektrodynamik hat das Potential noch mehr Freiheitsgrade. So kann für ein Potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> und das zugehörige Vektorpotential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {A}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/391292ffadc65b0cde3e96f23afcdb811619dd95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:3.009ex;" alt="{\displaystyle {\vec {A}}}" loading="lazy"></span> die folgende <a href="Eichtransformation" title="Eichtransformation">Eichtransformation</a>

</p>

<msup>

</msup>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi '({\vec {r}},t)=\varphi ({\vec {r}},t)-{\frac {\partial }{\partial t}}\Lambda ({\vec {r}},t)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6dbbc4497f8a5780bcedc54d7491a08887bd9a12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.988ex; height:5.509ex;" alt="{\displaystyle \varphi '({\vec {r}},t)=\varphi ({\vec {r}},t)-{\frac {\partial }{\partial t}}\Lambda ({\vec {r}},t)}" loading="lazy"></span></dd>

<msup>

<mover>

</mover>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}'({\vec {r}},t)={\vec {A}}({\vec {r}},t)+\operatorname {grad} \Lambda ({\vec {r}},t)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1af0880ffc0d318245818ffe345abce259d07ca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.357ex; height:3.843ex;" alt="{\displaystyle {\vec {A}}'({\vec {r}},t)={\vec {A}}({\vec {r}},t)+\operatorname {grad} \Lambda ({\vec {r}},t)}" loading="lazy"></span></dd></dl>

<p>durchgeführt werden, um ein neues Potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi '}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41699a838e24ad24212d37ff18eabb15a7765cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.205ex; height:3.009ex;" alt="{\displaystyle \varphi '}" loading="lazy"></span> und Vektorpotential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {A}}'}">

<msup>

<mover>

</mover>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}'}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58a91e9dd995ce444635c96402f18c9d9f8e40ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:3.343ex;" alt="{\displaystyle {\vec {A}}'}" loading="lazy"></span> zu erhalten, die dieselben elektrischen und magnetischen Felder erzeugen.

</p><p>Die beiden am häufigsten verwendeten Eichungen sind die <a href="Lorenz-Eichung" title="Lorenz-Eichung">Lorenz-Eichung</a> und die <a href="Coulomb-Eichung" title="Coulomb-Eichung">Coulomb-Eichung</a>. Es sind aber auch beliebig viele andere Eichungen möglich.

</p>

<div class="mw-heading mw-heading2"><h2 id="Messung_und_der_Zusammenhang_mit_der_elektrischen_Spannung">Messung und der Zusammenhang mit der elektrischen Spannung</h2></div>

<p>Das Potential eines elektrischen Feldes ist nicht eindeutig definiert, es kann immer eine beliebige Konstante dazu addiert werden, die von der Wahl des Nullpotentials abhängt (siehe <a href="Eichfreiheit" class="mw-redirect" title="Eichfreiheit">Eichfreiheit</a>). Der konkrete Wert des Potentials an einem Ort <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{0}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{0}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef4a62186256445a64da937f624fd1d892fad9ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{0}}" loading="lazy"></span> kann deshalb beliebig gewählt werden.

</p><p>Hingegen ist die Potential<i>differenz</i> zwischen zwei Punkten, auch <a href="Elektrische_Spannung" title="Elektrische Spannung">elektrische Spannung</a> genannt, eindeutig und kann deshalb auch gemessen werden.

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Elektrochemisches_Potential" title="Elektrochemisches Potential">Elektrochemisches Potential</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-:0-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:0_1-0">a</a></sup> <sup><a href="#cite_ref-:0_1-1">b</a></sup></span> <span class="reference-text"><a href="Wolfgang_Demtr%C3%B6der" title="Wolfgang Demtröder">Wolfgang Demtröder</a>: <cite style="font-style:italic">Experimentalphysik 2 Elektrizität und Optik</cite>. 7., korr. und erw. Auflage. Springer-Verlag GmbH, Berlin 2018, ISBN 978-3-662-55789-1.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Elektrisches+Potential&rft.au=Wolfgang+Demtr%C3%B6der&rft.btitle=Experimentalphysik+2+Elektrizit%C3%A4t+und+Optik&rft.date=2018&rft.edition=7.%2C+korr.+und+erw.+Auflage&rft.genre=book&rft.isbn=9783662557891&rft.place=Berlin&rft.pub=Springer-Verlag+GmbH" style="display:none"> </span></span>

</li>

<li id="cite_note-:1-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:1_2-0">a</a></sup> <sup><a href="#cite_ref-:1_2-1">b</a></sup></span> <span class="reference-text"><cite style="font-style:italic">Grundkurs Theoretische Physik 3 Elektrodynamik</cite>. 10. Aufl. 2013. Berlin, Heidelberg, ISBN 978-3-642-37905-5.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Elektrisches+Potential&rft.btitle=Grundkurs+Theoretische+Physik+3+Elektrodynamik&rft.edition=10.+Aufl.+2013&rft.genre=book&rft.isbn=9783642379055&rft.place=Berlin%2C+Heidelberg" style="display:none"> </span></span>

</li>

</ol>

<div>

Normdaten (Sachbegriff): <a href="Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="-print"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4128753-8">4128753-8</a></span> </div>

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